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# Acta Mathematica

## Volume 224 (2020)

### Number 1

### Purely unrectifiable metric spaces and perturbations of Lipschitz functions

Pages: 1 – 65

DOI: https://dx.doi.org/10.4310/ACTA.2020.v224.n1.a1

#### Author

#### Abstract

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of $\operatorname{Lip}_1 (X, m)$, the set of all bounded $1$-Lipschitz functions $f : X \to \mathbb{R}^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal{H}^n (f(S)) = 0$ is residual in $\operatorname{Lip}_1 (X, m)$. Conversely, if $E \subset X$ is $n$-rectifiable with $\mathcal{H}^n (E) \gt 0$, the set of all $f$ with $\mathcal{H}^n (f(E)) \gt 0$ is residual in $\operatorname{Lip}_1 (X, m)$.

These results provide a replacement for the Besicovitch–Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.

Received 16 January 2018

Received revised 26 March 2019

Accepted 5 February 2020

Published 31 March 2020